brachistochrone

Solves the brachistochrone problem with NeuralODE

Do you still remember the "quickest descent" problem we learned as a kid ? If not, please remind yourself a bit.

Here, we are going to solve the problem by minimizing the path parametrized by a neural network. The unusual thing is that our objective function involves an integration $$ t = \int_{x_0}^{x_1} \sqrt{\frac{1+(dy/dx)^2}{2 g (y_1- y_0) + v_0^2}} d x $$ We assume the particle moves from $(x_0, y_0)=(0, 0)$ to $(x_1, y_1)$ along the path $y(x)$ . And $g$ is the gravity constant, $v_0$ is the initial velocity.

Computing the objective function amounts to integrating an ordinary differential equation. And NeuralODE computes its gradient with respect to path, accurately and efficiently!

Play with the code and think about the following

• Change the value of $g$ and $v_0$. Does the solution agrees with your intuition ?
• What happens when $v_0\rightarrow 0$ ? Why does this happen ? Could you fix it ?
• Is there any other cool application you can thing of ? You are welcome to share with us.